If your champion have fear debuff it can prevent you from executing a skill

Makes perfect sense. I don't recall seeing the fear debuff icon in these cases, but that must be it because it explains exactly what's happening. Thanks JoinME.

I've very rarely encountered a few bugs with champions not doing anything and just missing their turn, though more in dungeons than arena iirc. Also visual bugs where champions DO attack but you don't actually see it happen. 

Or indeed it could be fear in which case it works as intended. 

I saw the answer concerning the "Fear" debuff which I fully understand, what has me baffeled is when I enter the arena the opposing team takes a full round of turns buffing up and then in some cases two full rounds of attack before I can do a single thing.  What is up with that?? Do some people have cheats that allow them multiple turns?

  BTW, I have a Progress Mission to acheive Silver Level in Arena.  LOL fat chance of that when you go into battle with a team who's power is 1/3 of yours and they just stand there and hammer you to death and you get no return chances.

Your team is likely very slow, speed 

There are many things that could make this happen, Bassagitator.

1. The speed of your champions is, like Trips allready mentioned, probably very low. You can increase it with artifacts.

2. The opponent's speed is probably verh high, so he starts.

3. The opposite team has probably champions, that can increase turnmeter and speed of their team (for example Apothecary).

4. There is an artifact set (Relentless), that gives a chance of 18% to get an extra round. 

5. There are champs that can reduce your turnmeter (for example Shatterbones: Attacks all enemies 3 times. Has a 50% chance to decrease the Turn Meter by 30%).

All this together can give them even more rounds before you start. It's not chess, where making a second move before the opponent did, would be cheating.

An easy way to get a faster team, is bringing the High Khatun to the Arena. She is an epic barbarian champion and daily reward at day 30. She brings a speedaura and an additional speedbuff for your team.